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Symbolic computation and the diffusion of shapes of triads

Published online by Cambridge University Press:  01 July 2016

Wilfrid S. Kendall
Wilfrid S. Kendall*
Affiliation:
University of Strathclyde
*
∗Present address: Department of Statistics, University of Warwick, Coventry CV4 7AL, UK.
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Abstract

This paper introduces the use of symbolic computation (also known as computer algebra) in stochastic analysis and particularly in the Itô calculus. Two related examples are considered: the Clifford-Green theorem on random Gaussian triangles, and a generalization of the D. G. Kendall theorem on the kinematics of shape.

The Clifford–Green theorem gives a remarkable characterization of the joint distribution of the squared-side-lengths of n independent Gaussian points in n-space, namely that this distribution is that of n independent exponential random variables conditioned to satisfy all the inequalities requisite if they are to arise as squared-side-lengths from a point-set in n-space. The D. G. Kendall theorem on the diffusion of shape identifies the statistics of the diffusion arising (under a time-change) as the shape of a triangle whose vertices diffuse by Brownian motion in 2-space or 3-space.

Symbolic Itô calculus is used to give a new proof of the Clifford-Green theorem, and to generalize the D. G. Kendall theorem to the case of triangles in higher-dimensional space whose vertices diffuse either according to Brownian motion or according to an Ornstein–Uhlenbeck process.

Keywords

SYMBOLIC MANIPULATIONCOMPUTER ALGEBRAREDUCESTATISTICS OF SHAPECLIFFORD-GREEN THEOREMSEMIMARTINGALEDIFFUSIONBROWNIAN MOTIONORNSTEIN–UHLENBECK PROCESSITO CALCULUSSTOCHASTIC INTEGRAL
Type
Research Article
Information
Advances in Applied Probability , Volume 20 , Issue 4 , December 1988 , pp. 775 - 797
Copyright
Copyright © Applied Probability Trust 1988 

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